The latus rectum of an ellipse is 10 and the minor axis Is equal to the distnace betweent the foci. The equation of the ellipse is

Options

A. `x ^(2) + 2y ^(2) =100`
B. `x ^(2) + sqrt2 y ^(2) =10`
C. `x ^(2) -2y ^(2) =100`
D. `sqrt2 x ^(2) + y ^(2) =10`

Answer

To find the equation of the ellipse given that the latus rectum is 10 and the minor axis is equal to the distance between the foci, we can follow these steps:

### Step 1: Understand the properties of the ellipse
The standard form of the equation of an ellipse is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The latus rectum \(L\) of the ellipse is given by the formula:

\[ L = \frac{2b^2}{a} \]

### Step 2: Use the given latus rectum
From the problem, we know that the latus rectum is 10. Therefore, we can set up the equation:

\[ \frac{2b^2}{a} = 10 \]

This simplifies to:

\[ b^2 = 5a \]

### Step 3: Relate the minor axis to the distance between the foci
The length of the minor axis is \(2b\) and the distance between the foci is given by \(2ae\), where \(e\) is the eccentricity defined as:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

Given that the minor axis is equal to the distance between the foci, we have:

\[ 2b = 2ae \]

This simplifies to:

\[ b = ae \]

### Step 4: Substitute the eccentricity
Substituting for \(e\):

\[ b = a\sqrt{1 - \frac{b^2}{a^2}} \]

### Step 5: Substitute \(b^2\) from earlier
From our earlier step, we have \(b^2 = 5a\). Substituting this into the equation gives:

\[ b = a\sqrt{1 - \frac{5a}{a^2}} = a\sqrt{1 - \frac{5}{a}} \]

### Step 6: Square both sides
Squaring both sides results in:

\[ b^2 = a^2\left(1 - \frac{5}{a}\right) \]

Substituting \(b^2 = 5a\) into this equation gives:

\[ 5a = a^2 - 5a \]

### Step 7: Solve for \(a\)
Rearranging gives:

\[ 10a = a^2 \]

Dividing both sides by \(a\) (assuming \(a \neq 0\)) gives:

\[ a = 10 \]

### Step 8: Find \(b^2\)
Substituting \(a = 10\) back into the equation \(b^2 = 5a\):

\[ b^2 = 5 \times 10 = 50 \]

### Step 9: Write the equation of the ellipse
Now we have \(a^2 = 100\) and \(b^2 = 50\). Therefore, the equation of the ellipse is:

\[ \frac{x^2}{100} + \frac{y^2}{50} = 1 \]

### Final Answer
The equation of the ellipse is:

\[ \frac{x^2}{100} + \frac{y^2}{50} = 1 \]